Some dark matter candidates are, according to theory, their own anti-particle. The only reason it is not a more explode-y space is that dark matter interacts very weakly with other matter, including itself, and therefore has not been identified yet.

That would make dark matter very lonely. If it interacts weakly, wouldn't there need to be more of it to account for the effects that dark matter was invented to explain?

Gravity is incredibly weak for individual particles. The reason we notice it in everyday life is because there are a lot of particles in the Earth pulling us the same way and all those little bits add up. This is the bit that we rely on to explain the galactic rotation curves (and to explain the cold spots in the CMB). If the dark matter only interacted gravitationally then it would almost impossible for us to make any direct detection of this sort (but it is also difficult to explain how so much of it was produced).

The idea of the WIMP is that the dark matter, in addition to interacting gravitationally, also interacts via another force called the weak force. While these interactions would have to be somewhat small so that the dark matter did not all explode, or collide too much with itself, it would still be much much stronger than the gravitational interactions on a per particle basis -- but would not "add up" the same way. [As a very simple analogy, the electric forces between protons and electrons are very strong compared to their gravitational attraction, but on large distances matter is almost neutral because opposite charges attract]

This idea is appealing to physicists because
1) if true, we have hope of detecting the dark matter and verifying its existence and
2) it tells us (very broadly) that we would produce the right amount of dark matter as the universe was cooling (the so-called WIMP miracle)

I didn't make the unqualified statement that only that particular frame was the correct one. My qualified statement was that this was the correct statement in terms of reference frames. That is, if you wanted to use relativity to make the statement in the article correct this is how you would do it.

Since one of the two statements I am making is that the majority of people who are making comments about relativity don't understand it, the "particularly" refers to people thinking that picking a different origin significantly alters the time taken [for these purposes the relatively small GR effect of the Earth's gravitational well makes near no difference]. For the people who say that it has been around 21 million years (implicitly assuming a frame that is comoving with the Earth) are indeed correct, but unless they were mentioning relativity somewhere else in their post they have given no evidence that they do or do not understand how time dilation works.

And by the way, if you think I'm being pedantic, I'm not. I'm objecting to your implication that anyone who didn't make that specific useless statement obviously doesn't understand relativity.

That is not the statement that I am making. The statement that I am making is that of the people that make statements about reference frames, the majority are incorrect, particularly those that make the statement that it has only been a couple of hours in the Earth's frame.

I am ambivalent about people who took the statement at the intended reading "discovered within hours of the first light rays arriving at Earth" as they have not demonstrated a lack of knowledge about relativity.

Many of the discussions are saying that an object 21 million years ago had to have blown up that long ago. There are even discussions of "there is no reference frame for time", suggesting that they are just using Galiean relativity. There are many other discussions pointing out incorrectly that in Earth's frame it was discovered within hours. The reason this is incorrect is that (to a good approximation) the star and Earth are not in relative motion, they are just separated by a considerable distance.

There have just been a couple of people pointing out the correct statement in terms of reference frames, namely that someone traveling at nearly the speed of light and following the beam would have experienced relatively little time. There are others that point out the statement actually means we detected the supernova within hours of the first rays of light reaching us.

I think the real reason people get infuriated with "nerds" is that they spend so much time insisting that they are right, being pedantic about the details while still harboring many misconceptions -- then getting defensive when these misconceptions are bought to their attention.

The electromagnetic and weak forces are combined into the electroweak force, but the theory predicts that in order to combine them (and be compatible with experiments) that you have to have an extra particle or particles to break the electroweak symmetry. That is the role of the Higgs. So our current theory of an electroweak description is accurate assuming that the Higgs or something that plays its role exists.

If there was no Higgs (or replacement) then we have a theory that still works phenomologically, but does not combine the electric and weak forces. Instead it has a bnuch of couplings that take very specific values *as if* the forces were combined.

Just to bolster the point that it does not do to be too narrow-minded, alchemy is in fact real. It is just that nowadays it goes under the much more respectable name "nuclear physics".

Although in fairness the people studying nuclear physics were interested in the pursuit of things unknown and ended up discovering that you could change one element into another, rather than being the (N+1)-th group to attempt to change one element into another by mixing various elements.

A Ph.D. tells you nothing except that the holder did some original research at an early point in their career.

There is also little if any correlation in being able to research, and being able to teach. Culturally, "everyone knows" the purpose of a phd is to become a professor and teach university students while collecting a $100K+ salary. The upper 50% to 10% cream of the crop actually get hired to do that. So, pretty much by definition, as a general cross section of the population, they are in the bottom of the barrel of teaching ability. So I'd be expecting, unless they're education phds, they're almost by definition probably not going to be good teachers.

[Emphasis added]

Sorry, by definition of what exactly? You assert that the upper 50% to 10% cream of the crop actually get hired [to teach university students while collecting $100K+]. Unless you are claiming that your criteria for determining the cream of the crop includes poor teaching ability (which I doubt you are) there is nothing by definition about your assertion that these people must be poorer than average teachers. I am guessing that you implicitly meant that cream of the crop referred to the research ability, and are then asserting (without proof or references) that research ability and teaching ability have a strong negative correlation. Which is particularly odd given your first sentence: the claim that there is little if any correlation between the two!

To put the argument more formally
Axiom #1) A Ph. D. tells you nothing except the holder did some original research [quoted from parent]
Axiom #2) There is little if any correlation between research ability and teaching ability [your first sentence]Your argument is that by definition [sic] we can put "someone has done research" together with "weak to no correlation between research and teaching" to conclude that Ph. D.s have the "bottom of the barrel teaching ability".

Your argument is contradictory. Your claim (Axiom #2) would lead us to say that we know nothing about the teaching ability of PhDs solely based on the fact they can do research. Maybe you meant that there was a correlation but that it was negative? In which case all you are doing is asserting a very strong (and certainly not by definition) statement into your argument.

If you are actually interested in looking at the correlation between research strength and teaching strength (something which is not captured soley by the definition, which is why they do research on it rather than just a formal proof) you could do worse than starting here:http://muse.jhu.edu/login?uri=/journals/journal_of_higher_education/v073/73.5marsh.pdf

If you just go to the first part of what he said - you can get more stuff for the same amount of money if you buy on a date that is slightly later than the date everyone else uses - it is completely rational. There is artificial demand in the market because everyone is attempting to do the same thing. It is perfectly rational to delay a few days to get more stuff -- or to spend less on Christmas and more on the kids on small things for the rest of the year. The *only* reason that the 25th is special is because religious people celebrate that particular day. It doesn't (and arguably shouldn't) have to occur on the same day as "present day". I see this as a reason to be mildly envious of his kids: they are learning rational behavior and budgeting skills.

As Roland went on to say, they found after trying this that the presents didn't actually matter. If that is a decision they have reached collectively then why should they care what some random stranger such as yourself may think? Not to be rude, but I think that I would worry about your kids more, feeling they have to please and conform.

It is correct to say that an accretion disk can form around a neutron star as well.

The distinguishing characteristic is that a neutron star bigger than its Schwarzschild radius. Not just a little bit bigger, but at least 11% bigger [see the Buchdahl-Bondi limit; this a theoretical limit for any perfect fluid spheres -- actual neutron stars don't come close to saturating that bound]. This means that the accretion of charged particles that is spiraling inward will eventually hit the surface, stopping the charged particles very rapidly. The radiation from suddenly stopping charged particles (Bremsstrahlung) is fairly distinctive, and is not seen here.

By contrast, an accretion disk around a black hole loses energy and eventually passes through the horizon. There is no sudden breaking and hence no Bremsstrahlung radiation It is the accretion disk and the lack of Bremsstrahlung that convinces us that the most likely candidate is a black hole.

[The reason the size limit was important is that as you get close to the horizon, redshifting makes things harder to see anyway. The point of the Buchdahl-Bondi theorem is that any perfect fluid sphere has to be about 11% bigger than the size of a black hole of equivalent mass. This limits the total redshift due to the object to a modest factor of 2, ensuring for a large class of matter (including neutron stars and all known matter to date) that the collision with the surface if it existed would be visible. This does not prevent unknown matter with exotic properties having s surface that is beyond the event horizon but close enough in the we would not see the Bremsstrahlung radition, but it is very difficult to construct "reasonable" solutions.]

I agree that you cannot simply throw money at a problem and hope that it just goes away. However I don't think that comparing public and private schools demonstrates the superiority of the private sector to redistribute wealth, at least not in the way that you think it does.

First, private schools are expensive. The parents sending their children to private school do not get to opt out of the taxes that support the public education system. Instead they are paying the government and the private sector. This shows these parents are willing to support their children's education; and having a home environment that understands the value of an education and is supportive of it is a really big piece of the puzzle.

Secondly the private schools are allowed to be more selective. You could argue, correctly, that this is a demonstration of the private system being more efficient: they will not accept known troublemakers, and discipline can be enforced more rigorously. The public school system is tasked with providing an education for the entire populace on the assumption that a society in which everyone is educated to at least a certain standard is beneficial to all. For example you want people who are voting for president to have the ability to think through the consequences and implications of the policies that person is advocating. We can argue the degree to which the public school system fails at its task, but it is quite different from that of the public school system. If we wanted a fair comparison we could look at research universities (both government and private) and compare them, as in both cases the institutions are trying to produce the best results possible and do not have to try a one-size-fits-all model.

I am certainly not debating that this particular initiative is successful, or that the education system is okay as it stands. However I do want to live in a society of people that have a basic knowledge of history, logic, mathematics and literacy. I think that is worth paying for. In particular your comment

Governments breed waste, inefficiency and tyranny and can never lead to a net gain for society when compared to a private institution.

does not seem to ring true. There is a reason why firefighters are not private institutions, after all!

But the fact that we can state the Uncertainty principle in a finite dimensional Hilbert space (as you point out) shows that the Uncertainty principle does rely on properties of infinity. It fact in the finite dimensional case it becomes somewhat easier to understand what is going on. Take the spin-1/2 system which is two dimensional. The eigenvectors any of the operators s_x, s_y or s_z form a basis for the state, however each operator's eigenbasis is not parallel to any other operator's eigenbasis. A vector which "lines up" like (1,0) and (0,1) in one basis cannot in the other two which we can draw on a sheet of paper (and remind students about breaking things into components).

In infinite dimensional cases things are more complicated because there are various subtitles that can arise. But these subtitles are not at the core of the uncertainty principle, merely a technical distraction that needs to be addressed.

And I really don't understand this statement:you need the features of Hilbert spaces that are unlike Euclidean spaces.All finite dimensional Euclidean spaces, for which we have a reasonable intuition, are Hilbert spaces. In the infinite dimensional case Hilbert spaces are defined to carry over the properties of Euclidean space while eliminating some of the perverse things that can happen in infinite cases (i.e. ensuring Cauchy sequences have limits in the space).

It really depends on what you think is relevant. For example if the purpose is to do research for people on Earth, you probably are interested (at least in part) in the time taken for the round trip, and how long people on Earth have to wait to see the benefits of their investment. If you are looking at colonization then you are probably more interested in the amount of time as experienced by the people travelling on the ship. In this case the difference between 99.9% of the speed of light, and 99.99% of the speed of light is significant.

To make the example concrete, let us take your example of Alpha Centuri:Distance: ~ 4 light-years.

99.9% of the speed of light:Time (Earth observer): 4 years and 1.5 daysGamma factor*: 22.4Time (Ship observer): 65 days

99.99% of the speed of light:Time (Earth observer): 4 years and part of a day.Gamma factor*: 70.7Time (Ship observer): 20.5 days.

So from the point of view of the *crew* the journey takes about a third the time, although from Earth you are correct in stating they are essentially the same.

* The gamma factor, or time dilation factor (or length contraction factor), is given by special relativity. If you speed is v and the speed of light is c thenGamma factor = 1/sqrt(1-(v/c)^2)

We could argue about whether or not numbers are anything other than concepts, but zero is a number. It is as much of a number as 1, 2, 3,......

One of the big hangups in early mathematics was that they were confused about the very thing you are: namely there is a difference between 0 and non-existence. I can ask what speed an object at rest has, and the answer is 0 (in whichever units you want). The answer is not "it does not have one" or that the speed does not exist. If you try doing velocity addition, or momentum conservation etc on an object at rest you will see that you do need to stick in '0' for the velocity.

(You may object -- there is no difference between an object not having momentum and thus not counting it, or having zero momentum. The easiest way to see a difference is to go to a difference reference frame -- then the momentums all transform. It would not make sense for a non-existant momentum to transform).

The real problem with division by zero is that
0 x = 0for any x. If we just have access to the RHS, we have no information about x, so asking "what do we need to multiply 0 by in order to get zero" the answer is "anything you want".

As to the question of whether or not infinity is a number, or simply a limiting process depends. There are not just one set of numbers, and you can actually define infinity to be a point, or simply a limit. (There are even multiple ways of "adding infinity" the most common of which in complex anaylsis is to add a "point at infinity" in which case -inf and +inf are the same thing.)

But it is true the most common way of dealing with numbers is to treat infinity as a limit.